Presentation on theme: "Introduction to PK/PD Modeling for Statisticians Part 1"— Presentation transcript:
1 Introduction to PK/PD Modeling for Statisticians Part 1 Alan Hartford, AbbVieASA Biopharm FDA-Industry Statistics WorkshopSeptember 16, 2015PKPD Modeling Training| Date 9/16/2015
2 Objectives and Outline for Part 1 Provide statisticians with the main concepts of PK/PD modeling (a.k.a., Pharmacometrics) (see outline below)Encourage statisticians to support pharmacometricians in their modeling effortsProvide an appreciation for PharmacometricsWe won’t be reviewing step by step modeling methods using AIC or p-valuesOutline: PK Basic Concepts, PK Compartmental Models, Technical Considerations, Software Considerations, Adding covariates to PK modelsPKPD Modeling Training| Date 9/16/2015
3 Introduction – PK and PD Pharmacokinetics is the study of what an organism does with a dose of a drugkinetics = motionAbsorbs, Distributes, Metabolizes, ExcretesPK EndpointsAUC, Cmax, Tmax, half-life (terminal), CminThe effect of the drug is assumed to be related to some measure of exposure. (AUC, Cmax, Cmin)Pharmacodynamics is the study of what the drug does to the body (dynamics = change)
4 PK/PD Modeling Procedure: Purpose: Fit a model to exposure data and estimate exposure at the same time points where we have PD data.Examine correlation between estimated exposure and PD (or other endpoints, e.g., AE rates).Might need to fit a mechanistic model with exposure data as explanatory variable and PD as response.Purpose:Estimate therapeutic windowDose selectionIdentify mechanism of actionModel probability of AE as function of exposure (and covariates)
5 Concentration of Drug as a Function of Time Model for Extra-vascular AbsorptionCmaxAUCConcentrationTmaxTime
6 Observed or Predicted PK? Exposure endpoints are not measured – only modeled, i.e., estimatedConcentration in blood or plasma is a biomarker for concentration at site of actionPK parameters are not directly measuredPKPD Modeling Training| Date 9/16/2015
7 Compartmental Modeling A person’s body is modeled with a system of differential equations, one equation for each “compartment”If each equation represents a specific organ or set of organs with similar perfusion rates, then called Physiologically Based PK (PBPK) modeling.The mean function f is a solution of this system of differential equations.Each equation in the system describes the flow of drug into and out of a specific compartment.
9 First-Order 1-Compartment Model (Intravenous injection) InputCentralVcEliminationk10Solution:
10 First-Order 1-Compartment Model (Intravenous injection) Parameterized with Clearance InputCentralAnother parameterization for the solution uses Clearance = Cl = k10 VcClearance = Volume of drug eliminated per unit timeVcEliminationk10Solution:
12 First-Order 1-Compartment Model (Extravascular Administration) Parameterized with Clearance InputkaCentralVcSolution:Eliminationk10F = Bioavailability(i.e., amount absorbed)
13 Parameterization ka, k10, V ka, Cl, V Micro constantka, Cl, VMacro constantNote that usually F, V, and Cl are not estimable (unless you perform studies with both IV and extravascular administration)Instead, apparent V (V/F) and apparent Cl (Cl/F) are estimated when only extravascular data are available
14 First-Order 2-Compartment Model (Intravenous Dose) Inputk12PeripheralCentralVc(Vp)k21Eliminationk10General form of solution:
15 First-Order 2-Compartment Model (Intravenous Dose) Parameterized in terms of“Micro constants”Note that including Vp over-parameterizes the model sinceInputk12PeripheralCentralVc(Vp)k21EliminationAc = Amount of drug in central compartmentAp = Amount of drug in peripheral compartmentk10
16 Web Demonstration(Requires installation of Adobe Shockwave player.)PKPD Modeling Training| Date 9/16/2015
17 Technical Considerations OutlineGeneral form of NLMEParameterizationError ModelsModel fitting(Approximate) Maximum LikelihoodFitting Algorithms
18 The Nonlinear Mixed Effects Model Pharmacokineticists use the term ”population” model when the model involves random effects.
20 Assay VariabilityAssays for measuring PK concentrations are validated for specific concentration ranges.If the concentration is higher than the upper limit of the validated assay, then the sample is diluted so the resulting diluted sample has PK concentration within the validated limits.If the concentration is lower than the lower limit of the validated assay, then the concentration is reported to be “below the limit of quantitation” (“BLOQ” or “<LLOQ”).
21 Assay Variability (cont.) The result of the assay specifications and the needed dilutions is that additional error is added into the measurement system.These errors can be accounted for in the statistical model.
22 Distribution of ErrorIn each case, the errors are assumed to be normally distributed with mean 0In PK literature, the variance is assumed to be constant (s2)Heteroscedastic variance is modeled using a proportional error termAnother option is to use the additive error model assuming a variance function R(q) where q is an m x 1 vector which can incorporate b, D and other parameters, e.g. R(q)=s2[f(b)]2, q=[s, bT]T
23 Error Models used for PK modeling Additive errorProportional errorAdditive and Proportional errorExponential errorPKPD Modeling Training| Date 9/16/2015
24 For the 1-compartment model parameterized with Cl, V, ka InputkaCentralVcEliminationk10And cov(logCli, logVi) is assumed to be 0 by definition of the pharmacokinetic parameters.
25 Maximum Likelihood Approach Is Used We obtain the maximum likelihood estimate by maximizingWhere p(yi) is the probability distribution function (pdf) of y where now we use the notation of yi as a vector of all responses for the ith subjectThe problem is that we don’t have this probability density function for y directly.
26 We use the following:Where p and p are normal probability density functions. Maximization is in =[bT, vech(D), vech(R)] TNotation: the vech function of a matrix is equal to a vector of the unique elements of the matrix.
28 Maximum LikelihoodGiven data yij, we use maximum likelihood to obtain parameters estimates for b, D, and s2. Because the mean function, f, is assumed to be nonlinear in bi in pharmacokinetics, least squares does not result in equivalent parameter estimates.
29 Approximate MethodsUse numerical approaches to approximate the integral and then maximize the approximationOptions:Approximate the integrand by something we can integrateFirst Order method (Taylor series)Approximate the whole integralLaplace’s approximation (second order approximation)Importance SamplingGaussian QuadratureUse Bayesian methodology
30 Algorithms UsedAvailable in NONMEMFirst Order First Order Conditional Estimation Laplace’s Approximation Importance Sampling Gaussian Quadrature Bayesian (Gibb’s Sampler; Not covered in this presentation)Approximate just the integrandOr approximate whole integral(NONMEM is the gold standard software package for PKPD modeling.)
31 First Order MethodApproximate with a first order Taylor series expansionIf the model assumesAnd Ri = s2I, then this is pretty straight-forward.You use a Taylor series expansion about bi.
32 Taylor Series Expansion With a first order Taylor series approximation expanded about b, the mean of the biLet this approximation beYou use this approximation in the integrand.
33 Substituting back in and simplifying … See slide 26.And now the exponent term is integrable and now we can maximize the likelihood.
34 Using Laplace’s Approximation A second order approximation can be constructed by using Laplace’s approximationIn this manner, the whole integral is approximated so no integration is needed.PKPD Modeling Training| Date 9/16/2015
35 Numerical Integration: Importance Sampling Consider a function g(b), thenTo get a numerical solution to the integral simply use a random number generator to sample many b from the distribution (b) and change the expectation to a sample mean.
36 Where h is the index for the sampling from p(bi). and
37 Problem!If each evaluation of the likelihood surface requires a resampling, then you introduce a randomness to your likelihood surface. The likelihood surface would have small perturbations which would affect your determination of a maximum. Solution: sample once and re-use this sample for each evaluation of the likelihood.
38 It turns out that importance sampling is not very efficient It turns out that importance sampling is not very efficient. To improve on this method, another method takes advantage of the normal assumption of distribution of bi.This method is called Gaussian Quadrature. Instead of a random sample, specific abscissas have been determined to best evaluate the integral.In particular, “adaptive Gaussian Quadrature” is a preferred method (not covered here).
39 Review of Approximate Methods First order: biased, only useful for getting starting values for better methods; converges often even if model is horrible. DON’T RELY ON THIS METHOD! Laplacian: numerically “cheap”, reasonably good fit Importance sampling: Need lots of abscissas, so not useful Gaussian Quadrature: GOLD STANDARD! But when data set large, method is slow and difficult to get convergence.
40 Additional NoteWhen your model does not converge, often it’s because you have the wrong model. Don’t switch algorithms just because of nonconvergence. First plot data and scrutinize choice of model.
41 SoftwareR – PKFIT package NONMEM (industry standard, 1979, FORTRAN) Monolix (Bayesian) WinBugs (PKBugs) Phoenix (windows program incorporating methods from NONMEM) SAS and R can be used to fit very simple PK models but, in general, not very usefulMerge this with earlier slidePKPD Modeling Training| Date 9/16/2015
42 Why is NONMEM the gold standard? Software needs easy input of PK models.Not many software packages allow for models written in terms of ODEs instead of closed form solution.More challenging for multiple dose settings.Functional form dependent on data.
43 Multiple Dose Model Daily Dose with Fast Elimination
44 Multiple Dose Model Daily Dose with Slower Elimination Super-position principle
45 Super-position Principle Assume dosing every 24 hoursAssume concentration for single dose isThen concentration, C(t) is
48 Modeling CovariatesAssumed: PK parameters vary with respect to a patient’s weight or age.Covariates can be added to the model in a secondary structure (hierarchical model).“Population Pharmacokinetics” refers specifically to these mixed effects models with covariates included in the secondary, hierarchical structure
49 Nonlinear Mixed Effects Model With secondary structure for covariates:
50 Part 1 Summary PK Basic Concepts (Cmax, Tmax, AUC …) PK Compartmental Models (derived mean function from differential equations)Technical Considerations (approximate maximum likelihood approach for fitting nonlinear mixed effects model)Software Considerations (many complications!)